Heteronuclear coherence transfer over a range of coupling constants. A broadband-INEPT experiment

JOURNAL

OF MAGNETIC

RESONANCE

69,264-282 (1986)

Heteronuclear CoherenceTransfer over a Rangeof Coupling Constants. A Broadband-INEPT Experiment STEPHEN WIMPERIS AND GEOFFREY BODENHAUSEN Institut de Chimie Organique, Universitt de Lausanne, Rue de la Barre 2, CH-1005 L.ausanne, Switzerland Received February 25, 1986 A procedure is described whereby the performance of NMR pulse experiments which

usefixed free precessionperiodswith durationsthat shouldLx.matchedto the reciprocal of a scalar, dipolar, or quadrupolar coupling constant or a resonance offset can be made less sensitive to the presenceof a rangeof coupling constantsor offsets.The example selected for full discussion is the INEPT experiment which uses lixed periods of free precession with durations that should ideally be chosen in proportion to the reciprocal of the heteronuclear scalar coupling constant. By considering an analogy with known composite pulses designed to compensate for radiofrequency field inhomogeneity AB, , a new experiment is derived, called “broadband-INERT,” which is effective over a wider range of coupling constants. This new method is shown, both by theory and experiment, to have a sensitivity advantage over the normal INEPT sequence in a variety of applications. Finally, the possibility is discussed of extending these desirable properties to many other NMR pulse. experiments. o 1986 academic RSS, IX. INTRODUCTION

A wide range of new methods in NMR involve pulse sequences that contain fixed periods of free precession (“delays”) with a duration r that must be matched to the reciprocal of the value of an internal spin interaction, such as a coupling or resonance offset. Typical examples of such experiments include the excitation of multiple-quantum coherence (1, 2), relayed magnetization transfer (3, 4), various heteronuclear coherence transfer methods (5, 6), creation of quadrupolar order of spin-l nuclei in anisotropic phase (7), and suppression of strong solvent lines by so-called “hard pulse” techniques (8, 9). The common purpose of such fixed delays is that a coherence present at the start of the delay must undergo a rotation through a specific angle 0 about a particular internal spin Hamiltonian, which for two-spin systems may be that of a homonuclear or heteronuclear scalar or dipolar coupling, or for single spins a quadrupolar coupling or simply a resonance offset: cZ;, = 27rJIkzII,

111

SF?&= 2nJI zS *

PI

8::

[31

= b& 1 - 3 cos%,)(31~Zz[Z - Ik * I])

&“g = b/J 1 - 3 cos%,)1& 0022-2364186 $3.00 Copyri@tt 0 1986 by Academic Press, Inc. All rigbt.3 of npmduction in any form reserved.

264

[41

BROADBAND

with bkl = ~yk7rh/(47&),

COHERENCE

TRANSFER

265

and s?sQ= (wo/3)(3SZ - P) 2:

= nr,

[61

where the symbols have their usual meanings (IO), and where the usual approximations have been made (weak scalar coupling, symmetric quadrupole tensor, etc.). An example of a rotation about an internal spin Hamiltonian occurs in the INEPT experiment which in its original form (5) does not include a rephasing interval after coherence transfer: Z 900-+18090-~~-9090 S

9090 Acquire

180

[71

where Z and S are two different spin-i nuclei, usually with +yr> ys, such as a proton and carbon-l 3. For simplicity, a system with only one nucleus of each species will first be considered. The internal spin Hamiltonian which describes the evolution of the Z spin coherence during the T delay is 27rJIS;. Using the notation of the product operator formalism (II) this evolution is described as -I, - e2Iz&

-I+os

8 + 2I$&sin 8

where B = TJT. For this experiment the required density operator at the end of the 7 delay is 2I&, since this term can be transferred by a 90W pulse applied to the I spins into -2I,S,, which is transformed subsequently by a 90W pulse on the S spins into -2Z& to enhance the observable S spin magnetization. The desired nominal rotation about 2Z& during T is therefore enorn= 7r/2. To obtain this rotation angle, a nominal coupling constant J,,, must be selected and the duration 7 fixed at (2J,,,)-‘. For optimum performance of the experiment J,,Omshould be an estimate, using either measurement, past experience, or guesswork, of the average coupling constant present. If within the spin system being studied, however, there are many IS pairs with a range of scalar coupling constants, then J,,, can at best only be an approximation to the true coupling constants. Any deviation of a true J from J,,, will result in a deviation of 6’ from 7r/2, and so the amplitude of 21,&, and hence that of the corresponding signals in the final INEPT spectrum, will be less than maximum. All NMR experiments which make use of rotations of this type suffer a degradation in performance if the true rotation is not the specific one desired. A similar problem occurs with simple radiofrequency pulses. These are normally used to achieve rotations in a three-dimensional operator space spanned by the orthogonal operators Z,, IY, and I,. Neglecting tilted fields due to resonance offsets, such rotations are described by a flip angle p and by a phase p, which defines the axis of rotation in the Z,, I,, plane. The inhomogeneity of the radiofrequency field A& gives rise to a range of flip angles 6, distributed about a nominal flip angle Bnom, which corresponds typically to the B, value in the center of the sample. In recent years, the use of composite pulses (12, 13) has become widespread in NMR to combat imperfections associated with inhomogeneous B, fields.

266

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Despite the attention lavished on the design of composite pulses to compensate for imperfections of radiofrequency pulses, much less attention has been directed toward compensation of errors associated with rotations about internal spin Hamihonians. Examples have been discussed of site-selective inversion sequences (“bilinear rotation decoupling”) (14, 15) and of site-selective excitation sequences (15, 16) compensated for a range of heteronuclear scalar coupling constants, and of solvent suppression sequences (8, 9) compensated for a range of resonance offsets. Barbara, Tycko, and Weitekamp (I 7) have shown that there is a formal analogy between the standard “excitation sandwich” used for creation of double-quantum coherence in a homonuclear two-spin-i or isolated spin- 1 system: Z or S

900-~~-18090-f~-90180

191

where 7 = (24-l or 7 = (7r/2wo), etc., and a conventional radiofrequency pulse of flip angle fl in the presence of A& inhomogeneity. Using this analogy they designed sequences for generation of double-quantum coherence compensated for a range of scalar, dipolar or quadrupolar coupling constants based on existing composite 90” pulses compensated for A& inhomogeneity. In this paper, it is shown how to compensate pulse sequences that involve a specific rotation about an internal spin Hamiltonian. This can be any of the Hamihonians of Eqs. [ l]-[6]. The theory, which loosely follows that. of Ref. (17) but in a different form, and the experiments derived from it are strictly speaking only applicable to systems with two Z = 1 spins or to isolated spins in the case of quadrupolar or offset Hamiltonians. However, it will be seen that there are many situations where, with care, compensated sequences may be applied to multispin systems. This paper will show the derivation and experimental verification of only one sequence in detail; an INEPT sequence compensated for a range of heteronuclear scalar coupling constants. This experiment will be referred to as a “broadband-INEPT” method, since the bandwidth with respect to the frequency domain of the scalar couplings is extended in comparison to that of the usual INEPT sequence. The derivation of this sequence has all the features needed to derive any of the other broadband sequences which are mentioned in the final section. In order that the INEPT sequence be discussed more easily it will be broken down into two parts; the “preparation sequence” and the “reconversion sequence.” This nomenclature is clarified in Fig. 1. The preparation sequence, where coherence evolves on the Z spin, is designed to create 2Z& order with maximum amplitude. The reconversion sequence, where coherence evolves on the S spin, may consist of a simple 90” pulse (as in INEPT without rephasing, see Eq. [7]), or,as shown in Fig. 1, the reconversion may consist of a sequence of pulses and delays designed to rephase the antiphase S spin coherence so that a decoupled signal may be acquired. THEORY

Compositerotations.The basic building block from which experiments designed to compensate for a range of heteronuclear scalar coupling constants can be constructed is the pulse sequence P(B,cp = 0) = 9orJ-7-90~~~

4-

Preparation

-,e.

Reconversion

and

Detection

__+

I

(8)

Normal

Normal

(b)

Broadband

I

Normal

I

S Normal

: s

180

:

Broadband

180

Cd)

;

FIG. 1. Pulse sequences for polarization enhancement of insensitive nuclei as discussed in the text. The conventional rephased INEPT sequence (a) is broken down into two parts: a normal preparation sequence of length 7 where coherence evolves on the Z spin, and a normal reconversion sequence of length 7’ where it evolves on the S spin. Schemes (b) and (c) show hybrid INEPT sequences-part normal and part broadband-and (d) shows the fully compensated INEPT sequence, broadband both in preparation and reconversion, with an overall length 37 + 37’.

267

268

WIMPERIS

AND

BODENHAUSEN

which, as will be shown below, leads to a convenient form of the corresponding propagator. The symbol P(B, (o) is a shorthand notation for this building block, where 0 = TJT and where the phase cp refers, by convention for this paper, to the phase of the first pulse (cp = 0 if the first pulse is applied along the x axis). Obviously, when this sequence is applied off-resonance, it is necessary to insert refocusing pulses in the center of the r delay to refocus chemical shifls while allowing the heteronuclear coupling to evolve undisturbed. For the derivation which follows, however, these pulses will initially be omitted since refocusing pulses can be inserted later into any new sequence derived. When applied to a heteronuclear scalar coupled two-spin-b system, the sequence of Eq. [lo] leads to a unitary transformation of the density operator 4

d = we, d4wmt

ill1

(0)

with the propagator U(0, cp = 0) = exp(+if?rZ&xp(-i82Z&)exp(-i&rZ,)

[I21

where 0 = TJT is the time variable. The constituent propagators appear as usual (II) in antichronological order. The propagator of Eq. [ 121 can be written in the simplified form (II): u(e, (0 = 0) = exp(-h92Z&. 1131 The propagator U(0, q~ = 0) transforms the set of density operator components Z,, 2Z,&, 2Z,&, in the following manner: Z, 2L&

e24.s e2zpz

&OS e + 2Zssin )

2Zs~0s

e

e

- Zain e

where the shorthand “arrow notation” of the product operator formalism (II) has been used, writing the argument of the exponential operator U-’ (without the imaginary i) above the arrows. These transformations imply that the effect of the propagator U(& cp = 0) can be viewed as a rotation in a three-dimensional space spanned by the orthogonal operators Z,, 2Z&, 2Z,& as long as the initial density operator itself lies within the same spin space, and as long as only a two-spin-i IS system is considered. The same applies to the phase-shifted analogs U(& 9): if the first pulse is applied along an axis cp, the propagator u(e, cp) = exp[-i8(2Z&cos

9 - 2Z$?$n p)]

]171

is obtained, again leading to rotations in a space spanned by the three operators Z,, 2Z,,&, and 2Z,+S’=. Broadband composite rotations. As described in the Introduction, pulse experiments often use r delays during which it is desired that the angle 0 = TJT takes a particular value (7r/2 in the case of conventional INEPT). Often, however, there is a range of true B’s reflecting a range of coupling constants J. As shown in Fig. 2, for the pulse sequence of Eq. [lo] such a range of B’s leads to a range of rotation angles distributed

BROADBAND

COHERENCE

TRANSFER

269

lW. 2. The action of the propagator U(0, p = 0) of FQ. [ 131 on the equilibrium density operator I, shown for a range of values of 0 = ?FJTdistributed about flaom= 90”.

about the nominal rotation angle c!&,~. If it is wished to compensate for a range of angles 6, then composite sequences P(I?, cp)P(B’,(p’)Z’(V’, cp”)- - *with rotation angles distributed about the nominal rotation angles finom, til,,, , &, - * - can be constructed in exact analogy to composite pulses designed to compensate for A& inhomogeneity h%4B’)d(B”)s * * * with flip angles distributed about the nominal flip angles &,,, &In, Kxn~ * *a This then is the key to compensating the INEPT preparation sequence. The basic unit, instead of &, is the analog of Eq. [lo] with variable phase cp: p(fl, d =

90~7+90,+l80

iI81

where +Jrefers, as before, to the phase of the first pulse. The composite sequence to be derived from this building block may be constructed in direct analogy to a A& inhomogeneity compensating 90” pulse. The composite sequence should be designed to rotate the density operator from Z, into the 2Z& 2Z$& plane, preferably with a well-defined phase (see below), over a range of values of 8. An Z spin 90” pulse with suitable phase is then applied to transform the antiphase coherence created by the composite sequence into 2Z&. The choice of a A&compensating 90” pulse on which to base the new INEPT sequence is crucial. Because the Z spin antiphase coherence created by the composite sequence has to be transformed by a pulse into 2Z&, it is necessary that this coherence

270

WIMPERIS

AND

BODENHAUSEN

have a e-independent phase in the 24&, 21,& plane. Most A&-compensating 90” pulses, for example the sequence (/3)0(&c, where fi is distributed about &,,,, = 90” (12), do not have this desirable property. Indeed, when applied to an initial density operator I,, the sequence (/3)&l), creates coherence in the Z,, I,, plane with a range of phases that are strongly dependent on the true flip angle /3. Such composite pulses would, when transformed into composite rotations in the spin space of Fig. 2, generate antiphase coherences in the ZZ$&, 21,& plane with a range of phases dependent on 0, and hence would be unsuitable for the INEPT application discussed here. However, some ABi-compensating 90” pulses have been proposed which produce a very small p-dependent phase shift when applied to I,: b31om3h2o (~/2)O(s)90(B)i80(ls/2),

(@0(28ho5(%h (3810(4P)169(2r8)33(28),78

[I91

WJI 1211 P4

where /3 is distributed about j3,,, = 90”. The first two sequences were proposed by Levitt (18) the latter two by Tycko et al. (19). All these pulses are suitable candidates on which to base the broadband version of the INEPT experiment, but the best sequence, in terms of distortion-free bandwidth and conciseness, appears to be the sequence G~)oW~~O of&. V91. Derivation of the broadband-INEPT preparation sequence. First the analog of the A&-compensating pulse (/3)o(2fl)lzo is constructed in the spin space of Fig. 2: 1 (900-7-90180)(90,20-27-90300)

]231

where T = (2&J’. The rotations achieved by the two constituent parts that are set in brackets in Eq. [23] are depicted in Fig. 3. The sequence creates I spin coherence in antiphase with respect to spin S with a phase cp = +60” with respect to the 21xS, axis, without any appreciable e-dependent phase shifts. This antiphase coherence can now be transformed into -21& order by the application of a simple 90, pulse along an orthogonal axis with cp = 150”. Thus the broadband INEPT preparation sequence with compensation for a range of heteronuclear scalar coupling constants, but as yet without refocusing pulses, is 1 900~7-901809012,3-27-903@,90,50

[241

where 7 = (2J,,,)-’ as before. The theoretical performance of this sequence for creating 2Z& order in a heteronuclear two-spin-f system compared to that of a conventional INEPT experiment can be calculated using product operators (I I). The normal INEPT preparation sequence creates 21& order with the 6 dependence: fnom(e) = -sin e

where B = by

7rJ7

t251

as usual. For the broadband preparation sequence of Eq. [24] it is given fiw

= t sin e cos 28 - f cos8 sin 28 - 3 sin 8.

WI

BROADBAND

COHERENCE

TRANSFER

271

FIG. 3. The action of the pulse sequence P(0, Q = O)p(20, q~ = 120) of Eq. [23], described by the sequence of propagators U(28, p = 12O)U(0, (p = 0), on the equilibrium density operator I,. Only one trajectory is shown for a single value of 0 = ~FJTwith 0 FJ 70”. Both rotations are positive (clockwise when looking along the rotation vector).

Figure 4 shows plots of the theoretical performance of the normal and broadband INEPT preparation sequences over a range of 13running from 0 to 180”. Clearly, the broadband sequence is more effective at creating 2Z& order when 0 # 90” than the conventional sequence. The reduced sequence. It is well known that adjacent pulses that occur in sequences like that of Eq. [24] (e.g., 901a-,90,20) can be simplified into single pulses that use only the more commonly available 90” phase shifts (I 7). This procedure is now applied in a slightly different form to the sequence of Eq. [24]. First the sequence of Eq. [24] is expanded using the equivalence 90V = @OO(O,, where cpzrepresents a positive rotation through an angle (o about the z axis: Z 9Oo-~-9O~8o12O~,9Oo12O~-2~-12O-,O,s~12O~15O~,9O~15O,.

t271

This can be simplified by combining the adjacent z rotations and by dropping the final z rotation completely since it does not a&ct the relevant 21,X, term created by the excitation sequence: z 900-T-9o*!jo 120-,900-27-90,

*030-r901J.

PI

The z rotations can now be expanded using the equivalence (o-, = 900~09090180: z 90~-7-90i*~90~12O9#-J90,&0~-27-90,*()90~30$X)90,~90(J.

v91

272

WIMPERIS

AND BODENHAUSEN

I 90

I 180'

0 FIG. 4. Theoretical efficiency (coefficient of longitudinal two-spin order 2I& as a function of 8 = TJT) of the normal INEPT preparation sequence (dashed curve, function&,@) of Eq. [25]), and efficiency of the broadband preparation sequences of Eqs. [24] and [3 l] (solid curve, functionf&,,,#) of ECq.[26]).

The adjacent 901s0 and 900 pulses may now be discarded giving the sequence Z 900-+ 120W-27-309,, .

1301

This reduced sequence is considerably simpler and hence less prone to experimental errors than that of Eq. [24]. The final broadband INEPT preparation sequence compensated for a range of heteronuclear scalar coupling constants, with refocusing pulses now inserted in the T delays is Z 900-fr-180W-~~-12090--7-18090--7-3090. s

180

180

1311

Note that compensation fails if the phases of the 180W pulses are chosen to be other than as they are given, whereas the phases of the 180” pulses with no specified phase may be chosen freely. The theoretical performance of this sequence in a two-spin-f system is of course again given by Eq. [26], illustrated by the curve of Fig. 4. Henceforth, the pulse sequence of Eq. [3 l] will be referred to as a broadband-INEPT preparation sequence, where “broadband” in this context refers to bandwidth with respect to J. A rephased broadband-INEPT. In most applications, antiphase coherence transferred from Z to S using the INEPT experiment is allowed to rephase after transfer so that a decoupled signal may be acquired. This rephased INEPT experiment (20) has an

BROADBAND

COHERENCE

TRANSFER

273

obvious sensitivity advantage over the version without decoupling. The pulse sequence used for conventional rephased INEPT is 180

Z 900-fr-180W-+9090 180

s

Decouple

90W-f ~‘- 180W-f7’-Acquire

~321

where r = 7’ = (2J,,,)-’ for a system with isolated IS groups. The rotation angle 8’ = TJT’ in the T’ delay is subject to the same fractional variations as the rotation angle 19= TJT in the r delay, so the rephased INEPT experiment should also gain by using a broadband reconversion sequence. At the beginning of the reconversion sequence, the IS system is in a state described by the density operator -2Z&. A 90” pulse must be applied to create S spin antiphase coherence, and the simplest reconversion and detection sequence is s

90150

[331

where the phase of the pulse has been chosen for later convenience. Then the inverse (21) of the broadband composite rotation P(0, cp = O)P(20, cp = 120), obtained by reversing the order of the sequence and phase shifting through 180”, namely P(28, cp = 3OO)P(B,(c = 180), may be applied in order to rotate the S spin antiphase coherence into enhanced Zeeman order S, with compensation for a range of heteronuclear scalar coupling constants: s 90,~~90~~-27’-90,*()90~~~-7’-90~ [341 where the 180” refocusing pulses have again been omitted, assuming for the time being that the S spin radiofrequency carrier is exactly on resonance. A pulse which may have arbitrary phase is then appended to the sequence to convert S, into transverse S spin coherence: s

90~~~90~@)-27’-90~~~90~~()-7’-90()90,~~.

[351

After reduction and insertion of refocusing pulses the reconversion and detection sequence becomes Z S

180

180

Decouple

30W-r’-18090-r’-12090-~r’-18090-~r’-Acquire.

[361

Thus the compensated reconversion sequence of Eq. [36] is the reverse in time of the preparation sequence with omission of the first pulse. This is yet another example of symmetric excitation and detection (I, 22). The two pulse sequences which have been derived, the broadband-INEPT preparation and reconversion sequences, do not have to be used in conjunction with one another. Many applications may require the use of the broadband-INEPT preparation sequence on its own or use of the broadband-INEPT reconversion sequence in conjunction with a conventional INEPT preparation sequence. These hybrid schemes are depicted in Figs. lb and c. Neither the derivation of the broadband-INEPT preparation sequence nor that of the broadband-INEPT reconversion sequence given above is unique. For the preparation sequence of Eq. [3 11, any A&-compensating 90” pulse that produces only small

274

WIMPERIS

AND

BODENHAUSEN

@dependent phase shifts when applied to an initial density operator Z, can be used as a starting point (see Eqs. [ 19]-[22]). Indeed even with the A&compensating (&(2&0 used above more than one equivalent sequence can be derived. For the reconversion sequence of Eq. [36] any A&compensating 90” pulse (not only the phase-distortionfree sequences of Eqs. [ 19]-[22]) can be used as a basis for the sequence. The use of a A&-compensating pulse that is not correcting for phase errors (e.g., (/?)0(2/3)90where j3 is distributed about /I,,,,, = 90“) as a basis for a broadband-INEPT experiment would however result in severe phase distortions as a function of J which would be impossible to correct with conventional linear offset-dependent phase-correction procedures. For most applications therefore the sequence of Eq. [36] is to be preferred. AREAS

OF

APPLICATION

The above derivations assume the existence of only isolated heteronuclear scalar coupled IS pairs within the system under investigation, where Z and S both have spin1. For the broadband-INEPT sequences to be of any practical use, however, they will have to be effective in spin systems of greater complexity. In the discussion below, areas where the preparation sequence of Eq. [3 l] might be useful are described separately from those where the reconversion sequence of Eq. [36] is likely to be fruitful. Applications of the broadband-INEPT preparation sequence. In organic chemistry the INEPT experiment is most frequently used to enhance either carbon- 13 or nitrogen15 signals using proton magnetization as a reservoir of polarization. If the one-bond heteronuclear coupling is used for obtaining the INEPT enhancement, the relevant spin groups are IS, I# or I$ (i.e., CH, CH2, and CH3 for carbon-13, or NH, NH2, and NH: for nitrogen-15). If the protons are magnetically equivalent, the proton satellite spectrum is a simple doublet and the broadband-INEPT preparation sequence should be effective at compensating for a range of one-bond heteronuclear coupling constants. However, the values of these coupling constants usually fall within a fairly narrow range, typically 120-200 Hz for carbon-l 3 and 65-95 Hz for nitrogen-l 5. This means that if J,,, is chosen for the middle of the range then the enhancement obtained by the broadband-INEPT preparation sequence will be only slightly greater than that obtained by the normal INEPT in the worst possible case. For example, if J,,, is chosen as 160 Hz for carbon- 13 but if there is a true J of 120 Hz in the system under investigation then 0 = ~JT = 67.5’ for this coupling; therefore fnom(8= 67.5”) = 0.92 andfbroad (0 = 67.5”) = 0.99 (see Eqs. [25], [26]) and so this represents only a 7% advantage of the broadband excitation sequence with respect to conventional INEPT. Homonuclear proton-proton couplings will tend to degrade this performance (although the possibility exists of removing these effects using bilinear rotation decoupling (14)), and so it would seem to make little sense to use broadband-INEPT in this application unless there is a very exceptional range of one-bond heteronuclear coupling constants present. Such systems are rare in organic chemistry but considerably more common in inorganic studies, where INEPT is frequently used to enhance the signals of nuclei with low gyromagnetic ratios. In many cases, it is necessary to use long-range heteronuclear couplings for the enhancement of insensitive nuclei such as carbon-l 3 or nitrogen-15, since these are often found in nonprotonated sites or, as in the case of nitrogen- 15, with such rapidly

BROADBAND

COHERENCE

TRANSFER

275

exchanging attached protons that the one-bond couplings cannot be observed. Longrange coupling constants have a large range of values, typically “JIs = O-l 5 Hz for II 3 2. However, any proton-proton coupling constants will also be of this magnitude, and it is the competition of homonuclear and long-range heteronuclear couplings that usually leads to disappointing results in long-range INEPT experiments (23). The broadband-INEPT preparation sequence is three times longer than the conventional INEPT preparation sequence, and hence more sensitive to the presence of homonuclear Z spin couplings. Thus the broadband-INEPT preparation sequence will probably be most useful for carbon- 13 or nitrogen- 15 enhancement through long-range couplings in spin systems where the protons have negligible homonuclear couplings. Two areas where the broadband-INEPT preparation sequence might be of more general applicability are in the enhancement of signals of nuclei with spin S > $ (24) and in the so-called “reverse” INEPT experiment (25,26) where polarization is transferred from a dilute to an abundant spin. These areas of application, however, will be better understood in terms of the discussion of the next section. Applications of the broadband-INEPT reconversion sequence. Rephasing of the S spin antiphase coherence created by an INEPT sequence after polarization transfer through one-bond heteronuclear couplings presents well known problems if the system contains a mixture of ZS, Z,S, and I$ groups. Assuming optimal excitation of 2Z,S, order from the Z spins, Burum and Ernst (20) have shown that the observable signal for each of the above spin groups at the end of the rephasing delay T’ is proportional to f(W, IS) = sin 0’ [371 f(O’, Z2S) = 2 sin Wcos 8’ = sin 28’

[381

f(W, I$)

1391

= 3 sin B’cos*B’ = 0.75(sin 0’ + sin 30’)

where 8’ = ~JT’. Thus the optimum rephasing delay is T’ = (2J,,,)-’ for ZS, 7’ = (4J,,0,,,)-’ for Z,S, and 7’ = (5J,,)-’ for Z$. This last rephasing delay is a compromise since the sin 8’ and sin 38’ components in Eq. [39] cannot be rephased simultaneously by a simple delay. An overall compromise value of T’ is required if signals arising from several types of spin group are to be observed. If only IS and Z2S spin groups are present, one must set T’ = (3J,,,)-’ in normal INEPT, whereas if I$ groups are present in addition, a better compromise value is T’ = 3( lOJ,,&-‘. Greater intensity of the decoupled S spin INEPT signals can be achieved by using the broadband-INEPT reconversion sequence of Eq. [36] since there are no homonuclear couplings JSSto interfere with its successful operation if S is a dilute nucleus. As implied in Eqs. [37]-[39], the rephasing of the S spin multiplets due to couplings to groups of equivalent Z spins can be treated as if these multiplets consisted of superpositions of doublets with different effective J couplings (namely J, 2J, and 3J, see Eqs. [37]-[ 391). Thus, for IS and Z2S spin groups the broadband-INEPT reconversion sequence can be used with the compromise value of the rephasing delay T’ = (3J,&‘, and if Z$ spin groups are present as well then it can be used with the value T’ = (4Jnom)-‘. Table 1 shows the signal intensities expected from each type of spin group where (i) a simple rephasing delay has been used, (ii) the broadband-INEPT reconversion sequence has been used, and (iii) hypothetical perfect rephasing has occurred. It can be seen that

276

WIMPERIS

AND BODENHAUSEN

TABLE 1 Peak Intensities of Various Spin Groups Obtained Using the INEPT Sequence with the Various Rephasing Methods Described in the Text’ Z&S IS and ZrS groups present

ZS, Z& and ZyT groups present

12s

Normal rephasing with 7’ = (3.Z)-’

0.87

0.87

Broadband mphasing with T’ = (3J))’

0.97

0.97

Hypothetical perfect rephasing

1.00

1.oo

IS

zzs

zjs

Normal rephasing with r’ = 3( lOJ)-’

0.81

0.95

0.84

Broadband rephasing with T’ = (4J)-’

0.88

1.oo

1.33

Hypothetical perfect rephasing

1.00

1.oo

1.50

’ Optimal performance of the preparation sequence and identical heteronuclear coupling constants are assumed for all spin groups.

the broadband reconversion sequence is predicted to produce greater signal intensity from all types of spin group than the conventional sequence, especially for Z3Sgroups. In spin systems where the Z spins have even greater multiplicity (e.g., 3’P(CH3)3 or 73Ge(CH3)4) then the broadband-INEPT reconversion sequence should show to even greater advantage. The above discussion has assumed that all one-bond heteronuclear coupling constants are equal and that the effects of long-range couplings during the rephasing can be ignored. If these restrictions are relaxed in the analysis then it can be shown that the broadband-INEPT reconversion sequence suffers no relative loss of performance with respect to the normal rephased INEPT. If long-range heteronuclear couplings are used to enhance the signals arising from nonprotonated sites, the rephasing of the S spin coherence is even more of a problem. In this case, each S spin nucleus is usually coupled to a great many Z spins with a large range of coupling constants “J Is. The usual procedure in this situation is to use a short rephasing delay T’ = (6”J,,,)to inhibit excessive dephasing by “passive” couplings (27). Because of the complexity of such spin systems, and because the signal results from the sum of coherences transferred from several different Z spins, the broadband-INEPT reconversion sequence has been found to have little advantage over the normal rephased INEPT, except in cases where there are only a few Z spins coupled to each S spin. As mentioned previously, the arguments above also apply to the use of the broadband-INEPT preparation sequence as a means of enhancing signals of nuclei with spin S B $. Here, the Z spin coherence evolves as a superposition of doublets with effective coupling constants J, 2J, 3J, . . . , etc. and a compromise value of the 7 delay has to be used with the normal INEPT preparation sequence. Use of the broadband-INEPT preparation sequence with transfer through one-bond couplings should lead to increased sensitivity. The same observation applies to “reverse” polarization transfer from S to Z spins in Z,S systems.

BROADBAND

COHERENCE

TRANSFER

277

The applications of the broadband-INEPT preparation and reconversion sequences discussed in this section have concentrated largely upon the conventional uses of INEPT; namely as a one-dimensional technique for enhancing signals of nuclei of low gyromagnetic ratio. The broadband-INEPT sequences can, of course, also be used with minor modifications in two-dimensional heteronuclear experiments such as shift correlation (28) and relayed spectroscopy (4). An obvious limitation of the broadband-INEPT sequences should perhaps be mentioned, as can be seen from Fig. 1 the broadband preparation and reconversion sequences are three times longer than their normal counterparts. Thus, if fast T2 relaxation makes it necessary to shorten the T and T’ delays of the normal INEPT sequence, then it would not be wise to attempt to use the broadband-INEPT sequences. EXPERIMENTAL

In the experimental verification of the broadband-INEPT preparation and reconversion sequences of Eqs. [31] and [36], all the 180” refocusing pulses used were in fact composite 180” pulses. Two different composite 180” pulses were used, chosen because they yield little phase distortion (19): (2/3)~,(2@),~~(2&,to compensate for A& inhomogeneity, and (&(3&&Q, to compensate for resonance offsets (in both cases, 0 is distributed about /3,, = 90”). To be specific, the 180g0 pulses of Eqs. [31] and [361 were wlac~ with either W)30V8h50(W)30 or (8)135(38)225@)135 depending on the particular application. Standard phase cycling was used for both the normal and broadband INEPT sequences. For the broadband preparation sequences, the 120” and 30” pulses were alternated in phase together in conjunction with addition and subtraction of the signals. The same effect can be obtained by alternating the phase of the first I spin pulse. All spectra were recorded at room temperature on a Bruker AM-400 spectrometer equipped with a process controller with digital phase shifters on both I and S channels. As a test of the compensation properties of both the broadband-INEPT preparation and reconversion sequences in a simple IS system the spectra of Fig. 5 were recorded. They show carbon-13 spectra of formic acid in DMSO-de, with rephasing in 7’ and proton decoupling, measured (a) with the conventional rephased INEPT experiment of Fig. la and (b) with the combined broadband-INEPT preparation and reconversion experiment of Fig. Id. The value of T = 7’ was incremented between successive experiments so that 0 = 8’ = nJcH7 went from 30” to 150” in steps of 10” to mimic the presence of a wide range of heteronuclear scalar coupling constants. The intensities of the peaks recorded with the normal rephased INEPT sequence vary according to f:,,(e) = sin28 whereas those recorded with the fully compensated sequence vary according to&&e) = [a sin B cos 20 - 1 cos 6 sin 20 - t sin 012. The agreement between theory and experiment is quite satisfactory. The performance of the broadband-INEPT reconversion sequence in rephasing S spin coherences arising from transfer via one-bond couplings ’ Jls in I,+!3systems with n = 1,2, or 3 is demonstrated in Fig. 6. The two spectra shown were recorded (i) with the conventional rephased INEPT experiment (T’ = 3(1OJ,,,)-‘) and (ii) with

278

: r WIMPEIUS

AND BODENHAUSEN ,..”

a

_’:

:

‘.., ‘_ ‘. ..

:

‘. ..

50

._ ‘.

,

30

‘.

h

70

90

I

1 110

130

150°

. b :

-I30

90

FIG. 5. Experimental verification of (a) the normal INEPT experiment with rephasing and decoupling of Fig. 1a, and (b) the fully compensated broadband-INEPT of Fig. 1d. The spectra show the carbon- 13 signal of formic acid HCOOH (isolated IS pan). The intervals 7 = 7 ‘, and hence the phase angles 0 = 0: have been incremented in each case as shown in order to mimic the presence of a range of heteronuclear coupling constants. The actual coupling constant is 22 1 Hz, and 7 = r’ was varied from 0.75 to 3.77 ms. The dotted lines indicate the theoretical forms of the corresponding amplitude functions, given by the squares of the functions&,,,(tY) andf-(8) in Eqs. [25] and [26], both normalized to the highest peak in (a). Note that the bandwidth in (b) is increased without any loss in amplitude for the matched condition B = 90’.

the hybrid experiment of Fig. lc with a broadband-INEPT reconversion sequence (7’ = (4&J’) used in conjunction with a conventional preparation sequence. The sample was 2-aminobutanoic acid, CH3CH2CH(NH2)COOH, in a mixture of 40 and DMSO-(I’, , and Jnomwas taken as 129 Hz. The figure shows that the signal intensity from each of the protonated carbon- 13 nuclei is enhanced using the broadband-INEF’T reconversion sequence, especially for the CH3 group where there is a 30% increase in the peak height. The use of the broadband reconversion sequence to rephase S spin coherences transferred via long-range heteronuclear couplings is demonstrated in Fig. 7. The two nitrogen-15 spectra shown were recorded as above, (i) with a conventional rephased INEPT sequence and (ii) with a broadband-INEPT reconversion sequence combined with a conventional preparation sequence. The spectra show the four decoupled nitrogen- 15 signals of caffeine, CsHi0N40T, in a mixture of formic acid and CDC13 . For both experiments 7’ was set for the best compromise value of (6”J,,,)-’ where ‘Jno,,, was estimated to be 7 Hz, although the actual coupling constants varied between 1

BROADBAND

a

b

COHERENCE

279

b’

C

CH

a

TRANSFER

CH3

CH2

a’

b

C’

b’

C

C’

FIG. 6. Carbon-l 3 sptxtra of the three protonated sites of 2-aminobutanoic acid, CH&H+ZH(NH$XDH. The spectrum on top left was recorded using the normal INERT sequence with T = (2&,,,-’ and 8 = 3( lOJ~,,,.J’, and the spectrum on top right was recorded using the broadband-INEPT reconversion sequence of Eq. [36] witb 7’ = (4.&J’ and with a normal preparation sequence with 7 = (2&J’ (see Fig. Ic). J,, was assumed to be 129 Hz; the actual ‘Ja coupling constants for the CH, CH2, and CHr groups are 144,129, and 127 Hz, respectively. The spectral width was 5680 Hz in both cases.The additional enhancements obtained using the broadband sequence were 3% for the CH and CH2 groups and 30% for the CHJ group (see expansions at bottom).

and 12 Hz. It can be seen that the broadband-INEPT reconversion sequence gives greater peak intensities than the conventional sequence. DISCUSSION

This paper has shown the derivation of a modified INEPT experiment that is effective over a large range of heteronuclear s&u coupling constants. The theoretical derivation assumed that the spin system of interest was composed only of isolated pairs of scalar coupled IS nuclei. However, it was demonstrated that the new broadband-INEPT sequences could still be used successfully in more complex spin systems. Obviously it is not expected that the experiments proposed in this paper wilI replace the conventional INEPT technique altogether. The broadband-INEPT sequence will be useful in systems

280

WIMPERIS

AND BODENHAIJSEN

0

a

bc

d

a’

b’c’

d’

FIG. 7. Nitrogen-l 5 spectra of caffeine. The spectrum on top left was recorded using the normal INEPT sequence and the spectrum on top right using the broadband-INEPT reconversion sequence with a normal preparation sequence (see Fig. lc). In both cases, 7’ = (6&J’ where J- was taken as 7 Hz. The true long-range proton-nitrogen coupling constants varied between 1 and 12 Ha. The spectral width was 7040 Hz. The spectra are the result of 160 acquisitions in 30 minutes. The assignment of peaks (c) and (d) is tentative, but does not affect the arguments of this paper. The additional enhancements obtained using the broadband sequence can be seen more clearly in the expansions at the bottom. The expansions of peaks (b) and (b’) are reduced in vertical size by a factor 4. The large intensity of this signal with respect to the other peaks arisesbecause this nitrogen-l 5 alone benefits from enhancements from two proton sites in the molecule.

where the coupling constants are unknown. The broadband preparation sequence should be useful in systems where competition of homo- and heteronuclear couplings seems a priori unlikely, whereas the broadband reconversion sequence is most promising where the S spin spectrum features doublets, triplets, and quartets. Furthermore, this work is an attempt to explore the possibility of compensating a wide variety of NMR experiments for variations in the strengths of internal spin Hamiltonians. As mentioned in the Introduction, the procedure applied in this paper to the INEPT experiment can be used to derive broadband forms of many different NMR experiments. The action of the pulse sequence (900-~-90,so), which is the basic building block of our procedure, can be described by a simple propagator U(0, cp = 0) for various internal spin Hamiltonians. For example: Heteronuclear

scalar coupled IS spin system: U(0, fp = 0) = exp(-i82Z&

where

6 = TJT.

r401

BROADBAND

Homonuclear

COHERENCE

281

TRANSFER

scalar coupled AX spin system: U(O, cp = 0) = exp(-i82Z&J

where

Spin S = 1 in anisotropic phase with quadrupolar U(0, cp = 0) = exp(-ifl$(3S;

- S2))

splitting where

0 = nJr. uQ:

8=

WQT.

Isolated spin at resonance offset Q: U(d, p = 0) = exp(-i9ZJ

where

6 = QT.

For each of these propagators a closed three-dimensional spin space can be defined in the manner of Fig. 2 and hence compensated sequences can be constructed based on existing composite pulses. Using arguments of this sort Garbow, Weitekamp, and Pines (Z4) have proposed a broadband site selective inversion sequence (“bilinear rotation decoupling”) for heteronuclear scalar coupled IS spin-4 systems, and Barbara, Tycko, and Weitekamp (17) have proposed broadband sequences for creation of double-quantum coherence in a variety of systems. By using the propagators of Eqs. [40]1431 in composite sequences, we may construct broadband pulse sequences for excitation of double- and zero-quantum coherence in heteronuclear IS spin-4 systems, for excitation of S spin multiple-quantum coherence in scalar coupled Z,S systems where S has noninteger spin, for excitation of zero-quantum coherence in homonuclear scalar coupled AX spin-j systems, for homonuclear coherence transfer in scalar coupled AX spin-$ systems, for excitation of quadrupolar order of spin S = 1 in anisotropic phase, and for suppression of solvent lines. Many other broadband sequences are also possible, of course; the above list is merely an appetizer. The design of pulse sequences which compensate for a range of strengths of some internal spin interaction is certainly a fascinating area. However, as is amply demonstrated by the broadband-INEPT experiment proposed in this paper, the true test of any new sequence derived will be its practical performance in multispin systems. There remains, therefore, much work to be done in this area. ACKNOWLEDGMENTS We are indebted to Dr. M. H. Levitt for providing a copy of Ref. (13) prior to publication. This research was supported in part by the Swiss National Science Foundation (Grants 2.445-0.84 and 2.925-0.85). REFERENCES 1. 2. 3. 4. 5. 6. 7.

D. P. WEITEKAMP, Adv. Mugs. Reson. 11, 111 (1983). G. B~DENHAUSEN, Prog. NMR Spectrosc. 14, 131(1981). G. W. EICH, G. B~DENHAUSEN, AND R. R. ERNST, .I. Am. Chem. Sot. 104,373l (1982). P. H. BOLTON AND G. B~DENHAUSEN, Chem. Phys. Lett. 89, 139 (1982). G. A. MORRIS AND R. FREEMAN, J. Am. Chem. Sot. 101,760 (1979). D. M. D~DDRELL, D. T. PEXQ AND M. R. BENDALL, J. Mugn. Reson. 48,323 (1982). J. JEENER AND P. BROECKAERT,Phys. Rev. 157,232 (1967). 8. V. SKLENAR AND Z. STAR-, J. Mugn. Reson. 50,495 (1982). 9. P. J. HORE, J. Magn. Reson. 55,283 (1983). 10. R. R. ERNST, G. B~DENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Oxford Univ. Press, in press.

282 II.

12. 13. 14. IS. 16.

17. 18. 19. 20.

21. 22. 23. 24. 25. 26. 27. 28.

WIMPERIS

AND

BODENHAUSEN

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